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arXiv:1808.09314 · Physical review. B./Physical review. B

Dirac composite fermions and emergent reflection symmetry about even-denominator filling fractions

Hart Goldman, Eduardo Fradkin

Published Oct 25, 2018• 27 citations
arXivHTMLPDF

Abstract

Motivated by the appearance of a ``reflection symmetry'' in transport experiments and the absence of statistical periodicity in relativistic quantum field theories, we propose a series of relativistic composite fermion theories for the compressible states appearing at filling fractions $\ensuremath{\nu}=1/2n$ in quantum Hall systems. These theories consist of electrically neutral Dirac fermions attached to $2n$ flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to $\ensuremath{\nu}=1/2n$, and we show that such states can be related by the observed reflection symmetry, at least at mean-field level. We further argue that the lowest Landau-level limit requires that the Dirac fermions be tuned to criticality, whether or not this symmetry extends to the compressible states themselves.

DOI
10.1103/physrevb.98.165137

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Motivated by the appearance of a ``reflection symmetry'' in transport experiments and the absence of statistical periodicity in relativistic quantum field theories, we propose a series of relativistic composite fermion theories for the compressible states appearing at filling fractions \ensuremathν=1/2n\ensuremath{\nu}=1/2n\ensuremathν=1/2n in quantum Hall systems. These theories consist of electrically neutral Dirac fermions attached to 2n2n2n flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to \ensuremathν=1/2n\ensuremath{\nu}=1/2n\ensuremathν=1/2n, and we show that such states can be related by the observed reflection symmetry, at least at mean-field level. We further argue that the lowest Landau-level limit requires that the Dirac fermions be tuned to criticality, whether or not this symmetry extends to the compressible states themselves.
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2\nu=1/2ν=1/2 corresponds to an effective time reversal transformation (i.e. {k⃗j}\{\vec{k}_j\}{kj​}→\rightarrow→{−k⃗j}\{-\vec{k}_j\}{−kj​}) for composite fermions, and discuss how this connects to the absence of 2kF2k_{\rm F}2kF​ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ\LambdaΛ levels (analogs of electronic Landau levels) we determine the Λ\LambdaΛ level spacing and find it to be very nearly independent of the Λ\LambdaΛ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1)n/(2pn-1)n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the v = 1/2 quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, C P. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n/(2n + 1) and (n + 1)/(2n + 1), which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, n and n + 1, is now mapped to the same half-integer filling factor n + 1/2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a C P -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.
A recent experimental study [Pan et al., arXiv:1902.10262] has shown that fractional quantum Hall effect gaps are essentially consistent with particle-hole symmetry in the lowest Landau level. Motivated by this result, we consider a clean two-dimensional electron system from the viewpoint of composite fermion mean-field theory. In this paper, we show that while the experiment is manifestly consistent with a Dirac composite fermion theory proposed recently by Son, it can equally well be explained within the framework of nonrelativistic composite fermions, first put forward by Halperin, Lee, and Read.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
We show that a modified version of Son's Dirac composite fermion theory proposed by Seiberg et al gives a candidate unified description of the gapped and gapless fractional quantum Hall states within a single Landau level. Our main tool is the successive application of three-dimensional dualities to partially filled Landau levels of composite fermions, which imply that this theory has a complicated landscape of gapped vacua and critical points. This construction is the Lagrangian, or effective field theory, analogue of the flux attachment procedure. The critical points exist at even denominator filling and are well-described by a Fermi surface for a weakly coupled composite fermion coupled to an abelian Chern-Simons theory. The gapped states include odd-denominator filling fraction states with an abelian Chern-Simons description which we show matches the one expected for hierarchy states, as well as non-abelian states at even-denominator filling that arise from pair instabilities of the composite fermion's Fermi surface.
Motivated by the particle-hole symmetry at the half-filled lowest Landau level, many articles have taken the point of view, following D. T. Son, that composite fermions are Dirac particles and their Fermi sea is akin to that occurring at the surface of a three dimensional topological insulator. This presents a paradox, because the microscopic theory of composite fermions, which treats composite fermions as non-relativistic particles, has been confirmed in great detail. The authors discuss what features of the Dirac-composite fermion view can be reconciled with the microscopic theory. They also show how particle-hole symmetry can emerge in the Chern-Simons formulation of composite fermions.